Velocity in body frame and inertial frame of the center of mass

Question

Ok so I have this really basic question: I want to reconcile the theorem of relative velocities, which expresses the velocity of a particle, in fact the center of mass of a rigid body, in an inertial frame when given in a moving, rotating frame, with the result which is obtained when CoM and origin of the moving frame are coincident points, moving together. This is the problem as I see it, maybe it is not well posed and please do point this out if this is the case.

So, from the theorem of relative velocities, having a fixed frame (F) Oxy and a moving frame (M) O'x'y' and a particle P, the velocity of the particle in the moving frame is expressed as sum of the velocity of the moving frame wrt the fixed frame, velocity of the particle wrt the moving frame (therefore in moving frame components?), and rotation of the moving frame wrt the fixed frame.

$\ \vec{v}_{P/F} = \vec{v}_{O'/F} + \vec{v}_{P/M} + \vec{\omega}_{M/F} \times \vec{r}_{O'P} $

Now, let's imagine O' coincides at every instant in time with the CoM; the moving frame is therefore a body frame (B).

$\ \vec{r}_{O'P} $ is clearly zero now, the last term disappears. The second term should also disappear since the CoM is never changing its position from O'.

So we end up with $\ \vec{v}_{P/F} = \vec{v}_{O'/F} $, which looks a bit like a tautology to me but I guess I am just tripping up.

My problem is reconciling this with the results of kinematics:

$\ \vec{v}_{P/F} = R \;\; \vec{v}_{P/B} $

where $\ R$ is the rotation matrix obtained through Euler angles transformations. So can this be resolved by saying that $\ \vec{v}_{O'/F} $ could be expressed in any other fixed (at least at a particular time) reference frame and therefore the transformation can be made by a simple Euler sequence?

Answer 1

I don't agree with your first equation because $\vec{v}_{P/M}$ isn't in the same coordinate frame as $\vec{v}_{O'/F}$, but rotated somewhat.

First we consider the general case, where P is not coincident with O' and is not fixed on the body.

• Position Kinematics

  $$ \vec{r}_{P/F} = \vec{r}_{O'/F} + \mathrm{R}\, \vec{r}_{O'P/M} \tag{1}$$

• Velocity Kinematics

  Direct differentiation of the terms above, with $\dot{\mathrm{R}} = \vec{\omega}_{O'/F} \times \mathrm{R}$, and $\vec{r}_{O'P/M}$ not fixed.

  $$ \vec{v}_{P/F} = \vec{v}_{O'/F} + \vec{\omega}_{O'/F} \times \mathrm{R}\, \vec{r}_{O'P/M} + \mathrm{R}\, \vec{v}_{O'P/M} \tag{2}$$

  This is almost your first equation if you use $\vec{r}_{O'P/F} = \mathrm{R}\, \vec{r}_{O'P/M}$, and $\vec{v}_{O'P/F} = \mathrm{R}\,\vec{v}_{O'P/M}$

  $$ \vec{v}_{P/F} = \vec{v}_{O'/F} + \vec{\omega}_{O'/F} \times \vec{r}_{O'P/F} + \vec{v}_{O'P/F} \tag{3}$$

Now consider P as fixed on the body, the above are

$$ \vec{v}_{P/F} = \vec{v}_{O'/F} + \vec{\omega}_{O'/F} \times \mathrm{R}\, \vec{r}_{O'P/M} \tag{4}$$ $$ \vec{v}_{P/F} = \vec{v}_{O'/F} + \vec{\omega}_{O'/F} \times \vec{r}_{O'P/F} \tag{5}$$

Or consider P as moving, but coincident to O'

$$ \vec{v}_{P/F} = \vec{v}_{O'/F} + \mathrm{R}\, \vec{v}_{O'P/M} \tag{6}$$

$$ \vec{v}_{P/F} = \vec{v}_{O'/F} + \vec{v}_{O'P/F} \tag{7}$$

The above two are identical equation, just a matter of what is more convenient to track.